
// Make newform 16.7.c.a in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_16_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_16_c_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_16_7_c_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_16_7_c_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_16_c();"
function MakeCharacter_16_c()
    N := 16;
    order := 2;
    char_gens := [15, 5];
    v := [1, 2];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_16_c_Hecke();"
function MakeCharacter_16_c_Hecke(Kf)
    N := 16;
    order := 2;
    char_gens := [15, 5];
    char_values := [[-1], [1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 7;
    raw_aps := [[0], [0], [234], [0], [0], [-4070], [-990], [0], [0], [-31878], [0], [-55510], [-84942], [0], [0], [-29430], [0], [234938], [0], [0], [427570], [0], [0], [1378962], [-1472510], [-1702998], [0], [0], [458458], [1122210], [0], [0], [-2563470], [0], [5732298], [0], [-7657990], [0], [0], [6307290], [0], [3679882], [0], [14372930], [-13912470], [0], [0], [0], [0], [16814122], [-24941070], [0], [201058], [0], [-31596030], [0], [-27329382], [0], [40776010], [10828818], [0], [38391210], [0], [0], [-14570350], [41845050], [0], [76372450], [0], [4663802], [-77097150], [0], [0], [63187850], [0], [0], [-117587862], [-32712550], [-123211998], [-85770542], [0], [-30644822], [0], [-138726430], [0], [0], [78623298], [81560050], [-190571238], [0], [0], [0], [0], [0], [0], [0], [-60111558], [280651122], [0], [-281610758], [0], [-271209510], [0], [350121618], [0], [-372269950], [0], [-176743710], [0], [-144142702], [0], [78568490], [-230573070], [0], [0], [305573058], [0], [0], [555866010], [0], [90612938], [-589732990], [-604142550], [0], [0], [-287743302], [642174442], [0], [0], [711417850], [0], [0], [0], [360625930], [135039762], [-816390718], [639789930], [0], [747614970], [-517457358], [0], [-1103352822], [0], [0], [-602782022], [0], [-826038070], [856602450], [0], [0], [458387930], [-1316473182], [0], [0], [0], [0], [0], [-650820222], [1053073010], [-681896358], [0], [1490424210], [0], [0], [1339089570], [0], [0], [814875050]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_16_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_16_7_c_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_16_7_c_a(:prec:=1)
    chi := MakeCharacter_16_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 7));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_16_7_c_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_16_7_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_16_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,7,sign)));
    Vf := Kernel([<3,R![0, 1]>],Snew);
    return Vf;
end function;